Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier-Stokes equations
Linear and non-linear high order accurate residual distribution schemes for the discretization of the steady compressible Navier-Stokes equations
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DOI：
10.1016/j.jcp.2014.11.031
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发表时间：
2015-02
期刊：
影响因子：
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通讯作者：
R. Abgrall;D. Santis
中科院分区：
文献类型：
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作者：
R. Abgrall;D. Santis
提出了一种鲁棒的高阶精度残差分布（RD）格式，用于定常Navier-Stokes方程的离散。所提出的方法是非常灵活的：它是制定非结构化网格，无论元素的形状和空间维度的数量。采用解的连续近似，并使用标准拉格朗日形函数来构造离散空间，如有限元方法。设计RD格式的传统方法是：对任意单元求出总残差，将其分解为节点残差，并将其发送到单元的自由度，求解已组装的非线性系统，然后逐步收敛。该文件所解决的主要问题是，该技术依赖于在深度上的正常通量的连续性跨越元素边界：这不再是真的，因为梯度的状态解决方案出现在通量，因此连续性丢失时，使用标准的有限元近似。简单的解决方法导致非常差的准确性。为了科普数值解梯度的法向分量在单元表面上不连续的事实，在网格的每个自由度处恢复数值解梯度的连续近似，然后用相同的形状插值用于解决方案的函数，保持该方法的最佳精度。构造了线性和非线性格式，并用制造解的方法检验了它们的精度。数值方法也用于离散化的光滑和冲击层流在两个和三个空间维度。
A robust and high order accurate Residual Distribution (RD) scheme for the discretization of the steady Navier–Stokes equations is presented. The proposed method is very flexible: it is formulated for unstructured grids, regardless the shape of the elements and the number of spatial dimensions. A continuous approximation of the solution is adopted and standard Lagrangian shape functions are used to construct the discrete space, as in Finite Element methods. The traditional technique for designing RD schemes is adopted: evaluate, for any element, a total residual, split it into nodal residuals sent to the degrees of freedom of the element, solve the non-linear system that has been assembled and then iterate up to convergence. The main issue addressed by the paper is that the technique relies in depth on the continuity of the normal flux across the element boundaries: this is no longer true since the gradient of the state solution appears in the flux, hence continuity is lost when using standard finite element approximations. Naive solution methods lead to very poor accuracy. To cope with the fact that the normal component of the gradient of the numerical solution is discontinuous across the faces of the elements, a continuous approximation of the gradient of the numerical solution is recovered at each degree of freedom of the grid and then interpolated with the same shape functions used for the solution, preserving the optimal accuracy of the method. Linear and non-linear schemes are constructed, and their accuracy is tested with the method of the manufactured solutions. The numerical method is also used for the discretization of smooth and shocked laminar flows in two and three spatial dimensions.